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Lill's Method
In mathematics, Lill's method is a visual method of finding the real roots of a univariate polynomial of any degree. It was developed by Austrian engineer Eduard Lill in 1867. A later paper by Lill dealt with the problem of complex roots. Lill's method involves drawing a path of straight line segments making right angles, with lengths equal to the coefficients of the polynomial. The roots of the polynomial can then be found as the slopes of other right-angle paths, also connecting the start to the terminus, but with vertices on the lines of the first path. Description of the method To employ the method a diagram is drawn starting at the origin. A line segment is drawn rightwards by the magnitude of the first coefficient (the coefficient of the highest-power term) (so that with a negative coefficient the segment will end left of the origin). From the end of the first segment another segment is drawn upwards by the magnitude of the second coefficient, then left by the magnitud ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lill Method Quartic Example
Lill is a surname. When borne by Estonian individuals, it means "flower". People with the surname Lill include: * Alfred John Lill, Jr. (1880-1956), American former president of the Amateur Athletic Union and member of the United States Olympic Committee * Alick Lill (1904–1987), Australian rules footballer * Andreas Lill (born 1965), German drummer (Vanden Plas) * Anne Lill (born 1946), Estonian classical philologist and translator * Darren Lill (born 1982), South African racing cyclist * Denis Lill (born 1942), British actor * Eduard Lill (1830–1900), Austrian engineer and army officer * Erkki Lill (born 1968), Estonian curler and curling coach * Harri Lill (born 1991), Estonian curler * Heino Lill (born 1944), Estonian basketball coach and basketball player * Ivo Lill (born 1953), Estonian glass artist * Jim Lill (born 19??), American country musician * John Lill (born 1933), Australian cricketer * John Lill (born 1944), British classical pianist * Kristiine Lill (born 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Synthetic Division
In algebra, synthetic division is a method for manually performing Euclidean division of polynomials, with less writing and fewer calculations than long division. It is mostly taught for division by linear monic polynomials (known as the Ruffini's rule), but the method can be generalized to division by any polynomial. The advantages of synthetic division are that it allows one to calculate without writing variables, it uses few calculations, and it takes significantly less space on paper than long division. Also, the subtractions in long division are converted to additions by switching the signs at the very beginning, helping to prevent sign errors. Regular synthetic division The first example is synthetic division with only a monic linear denominator x-a. :\frac The numerator can be written as p(x) = x^3 - 12x^2 + 0x - 42 . The zero of the denominator g(x) is 3. The coefficients of p(x) are arranged as follows, with the zero of g(x) on the left: :\begin \begin \\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carlyle Circle
In mathematics, a Carlyle circle (named for Thomas Carlyle) is a certain circle in a coordinate plane associated with a quadratic equation. The circle has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the horizontal axis. Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons. Definition Given the quadratic equation :''x''2 − ''sx'' + ''p'' = 0 the circle in the coordinate plane having the line segment joining the points ''A''(0, 1) and ''B''(''s'', ''p'') as a diameter is called the Carlyle circle of the quadratic equation.E. John Hornsby, Jr.''Geometrical and Graphical Solutions of Quadratic Equations'' The College Mathematics Journal, Vol. 21, No. 5 (Nov., 1990), pp. 362–369JSTORJSTOR Defining property The defining property of the Carlyle circle can be established thus: the equation of the circle having the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paper Folding
) is the Japanese art of paper folding. In modern usage, the word "origami" is often used as an inclusive term for all folding practices, regardless of their culture of origin. The goal is to transform a flat square sheet of paper into a finished sculpture through folding and sculpting techniques. Modern origami practitioners generally discourage the use of cuts, glue, or markings on the paper. Origami folders often use the Japanese word ' to refer to designs which use cuts. On the other hand, in the detailed Japanese classification, origami is divided into stylized ceremonial origami (儀礼折り紙, ''girei origami'') and recreational origami (遊戯折り紙, ''yūgi origami''), and only recreational origami is generally recognized as origami. In Japan, ceremonial origami is generally called "origata" ( :ja:折形) to distinguish it from recreational origami. The term "origata" is one of the old terms for origami. The small number of basic origami folds can be combine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Margherita Piazzola Beloch
Margherita Beloch Piazzolla (12 July 1879, in Frascati – 28 September 1976, in Rome) was an Italian mathematician who worked in algebraic geometry, algebraic topology and photogrammetry. Biography Beloch was the daughter of the German historian Karl Julius Beloch, who taught ancient history for 50 years at Sapienza University of Rome, and American Bella Bailey. Beloch studied mathematics at the Sapienza University of Rome and wrote her undergraduate thesis under the supervision of Guido Castelnuovo. She received her degree in 1908 with Laude and "dignità di stampa" which means that her work was worthy of publication, and in fact her thesis "Sulle trasformazioni birazionali nello spazio" (On Birational Transformations in Space) was published in the Annali di Matematica Pura ed Applicata. Guido Castelnuovo was very impressed with her talent and offered her the position of assistant which Margherita took and held until 1919, when she moved to Pavia. In 1920 she moved to P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lill Method Folding Example
Lill is a surname. When borne by Estonian individuals, it means "flower". People with the surname Lill include: * Alfred John Lill, Jr. (1880-1956), American former president of the Amateur Athletic Union and member of the United States Olympic Committee * Alick Lill (1904–1987), Australian rules footballer * Andreas Lill (born 1965), German drummer (Vanden Plas) * Anne Lill (born 1946), Estonian classical philologist and translator * Darren Lill (born 1982), South African racing cyclist * Denis Lill (born 1942), British actor * Eduard Lill (1830–1900), Austrian engineer and army officer * Erkki Lill (born 1968), Estonian curler and curling coach * Harri Lill (born 1991), Estonian curler * Heino Lill (born 1944), Estonian basketball coach and basketball player * Ivo Lill (born 1953), Estonian glass artist * Jim Lill (born 19??), American country musician * John Lill (born 1933), Australian cricketer * John Lill (born 1944), British classical pianist * Kristiine Lill (born 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right Triangle
A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right angle (that is, a 90-degree angle), i.e., in which two sides are perpendicular. The relation between the sides and other angles of the right triangle is the basis for trigonometry. The side opposite to the right angle is called the ''hypotenuse'' (side ''c'' in the figure). The sides adjacent to the right angle are called ''legs'' (or ''catheti'', singular: ''cathetus''). Side ''a'' may be identified as the side ''adjacent to angle B'' and ''opposed to'' (or ''opposite'') ''angle A'', while side ''b'' is the side ''adjacent to angle A'' and ''opposed to angle B''. If the lengths of all three sides of a right triangle are integers, the triangle is said to be a Pythagorean triangle and its side lengths are collectively known as a ''Pythagor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thales's Theorem
In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's '' Elements''. It is generally attributed to Thales of Miletus, but it is sometimes attributed to Pythagoras. History There is nothing extant of the writing of Thales. Work done in ancient Greece tended to be attributed to men of wisdom without respect to all the individuals involved in any particular intellectual constructions; this is true of Pythagoras especially. Attribution did tend to occur at a later time. Reference to Thales was made by Proclus, and by Diogenes Laërtius documenting Pamphila's statement that Thales "was the first to inscribe in a circle a right-angle triangle". Babylonian mathematicians knew this for special cases before Thales proved it. It is be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lill Method Quadratic Example
Lill is a surname. When borne by Estonian individuals, it means "flower". People with the surname Lill include: * Alfred John Lill, Jr. (1880-1956), American former president of the Amateur Athletic Union and member of the United States Olympic Committee * Alick Lill (1904–1987), Australian rules footballer * Andreas Lill (born 1965), German drummer (Vanden Plas) * Anne Lill (born 1946), Estonian classical philologist and translator * Darren Lill (born 1982), South African racing cyclist * Denis Lill (born 1942), British actor * Eduard Lill (1830–1900), Austrian engineer and army officer * Erkki Lill (born 1968), Estonian curler and curling coach * Harri Lill (born 1991), Estonian curler * Heino Lill (born 1944), Estonian basketball coach and basketball player * Ivo Lill (born 1953), Estonian glass artist * Jim Lill (born 19??), American country musician * John Lill (born 1933), Australian cricketer * John Lill (born 1944), British classical pianist * Kristiine Lill (born 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicative Inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a rational number, fraction ''a''/''b'' is ''b''/''a''. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The reciprocal function, the Function (mathematics), function ''f''(''x'') that maps ''x'' to 1/''x'', is one of the simplest examples of a function which is its own inverse (an Involution (mathematics), involution). Multiplying by a number is the same as Division (mathematics), dividing by its reciprocal and vice versa. For example, multiplication by 4/5 (or 0.8) will give the same result as division by 5/4 (or 1.25). Therefore, multiplication by a number followed by multiplication by its reciprocal yiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reciprocal Polynomial
In algebra, given a polynomial :p(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n, with coefficients from an arbitrary field, its reciprocal polynomial or reflected polynomial,* denoted by or , is the polynomial :p^*(x) = a_n + a_x + \cdots + a_0x^n = x^n p(x^). That is, the coefficients of are the coefficients of in reverse order. They arise naturally in linear algebra as the characteristic polynomial of the inverse of a matrix. In the special case where the field is the complex numbers, when :p(z) = a_0 + a_1z + a_2z^2 + \cdots + a_nz^n, the conjugate reciprocal polynomial, denoted , is defined by, :p^(z) = \overline + \overlinez + \cdots + \overlinez^n = z^n\overline, where \overline denotes the complex conjugate of a_i, and is also called the reciprocal polynomial when no confusion can arise. A polynomial is called self-reciprocal or palindromic if . The coefficients of a self-reciprocal polynomial satisfy for all . Properties Reciprocal polynomials have several connec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
